Can a parrot and cockiteal share a cage

It depends on the personalities of the individual birds involved and other factors, such as sufficient cage space, sex of the birds, etc.  Each parrot has its own personality just like humans do.  Some get along together and some don't.  It's a chance we take when we own birds.  

In my 20+ years of experience breeding/raising keets and tiels, keets sometimes pick on tiels.  I've seen them ride on the tails of tiels, pick at their feet, climb on their backs, and other annoying behaviors.  This results in stress for the tiels, which can affect their immune system, making them susceptible to illness/disease.

I think it's not possible that a parrot and cockatiel share a cage.